Problem 52 - Elementary and Intermediate Algebra

Question

Factor. See Example 7 or Example \(12 .\) $$2 b^{2}-20 b+18$$

Step-by-Step Solution

Answer

The factored form is \(2(b - 1)(b - 9)\).

Step 1: Identify the Greatest Common Factor (GCF)

First, look for the greatest common factor (GCF) of all the terms. The coefficients are 2, -20, and 18. The GCF of these numbers is 2. Therefore, factor out the 2 from the expression: \[2(b^2 - 10b + 9)\]

Step 2: Factor the Quadratic Inside the Parentheses

Now, focus on factoring the quadratic expression inside the parentheses, \(b^2 - 10b + 9\). Look for two numbers that multiply to +9 (the constant term) and add to -10 (the coefficient of the linear term). These numbers are -1 and -9.

Step 3: Rewrite the Quadratic as a Product of Binomials

Use the numbers found in Step 2 to rewrite the quadratic as a product of two binomials. So, \(b^2 - 10b + 9\) can be written as: \[(b - 1)(b - 9)\]

Step 4: Combine with the GCF

Now, reintroduce the GCF factored out earlier to express the original quadratic as the product of the GCF and the two binomials: \[2(b - 1)(b - 9)\]

Key Concepts

Understanding the Greatest Common Factor

The greatest common factor, often abbreviated as GCF, is essential when simplifying expressions involving multiple terms. It represents the largest factor shared by all the coefficients in an expression. For instance, in the expression \(2b^2 - 20b + 18\), the coefficients are 2, -20, and 18. To determine the GCF, consider the factors of each coefficient.

• The factors of 2 are 1 and 2.
• The factors of -20 are 1, 2, 4, 5, 10, and 20.
• The factors of 18 are 1, 2, 3, 6, 9, and 18.

The common factor among these is 2, which makes it the GCF. By factoring out the GCF, you simplify the expression and make the next steps easier. Factoring out the GCF from the example expression gives us \(2(b^2 - 10b + 9)\). This simplifies the process of factoring further by reducing the size of the numbers involved.

Decomposing Quadratic Expressions

Quadratic expressions come in the standard form \(ax^2 + bx + c\). In the expression \(b^2 - 10b + 9\), our task is to factor it into a more manageable form. This process involves finding two numbers that multiply to the constant term and add up to the linear coefficient.For the quadratic \(b^2 - 10b + 9\):

• The constant term is +9.
• The linear coefficient is -10.

We need to identify numbers that multiply to +9 and add to -10.

• The chosen numbers for this quadratic are -1 and -9.

By using these numbers, we can rewrite the quadratic expression as a product of two binomials, which makes the expression easier to solve or simplify further.

Mastering Binomial Factorization

Once we have the suitable numbers from the quadratic expression, binomial factorization involves expressing the quadratic as a product of two binomials. From the expression \(b^2 - 10b + 9\), we identified -1 and -9 as the numbers that satisfy the criteria of our previous analysis.These numbers allow us to decompose the quadratic as:\[(b - 1)(b - 9)\]This step is crucial because it reduces the quadratic into two simpler expressions that are easier to interpret and use in further mathematical computations. Remember to include the GCF if you factored it out earlier. Hence, the complete factored form of the original expression \(2b^2 - 20b + 18\) is:\[2(b - 1)(b - 9)\]This process not only simplifies expressions but also lays the groundwork for solving polynomial equations efficiently.